On the Nagumo uniqueness theorem

By a convenient reparametrisation of the integral curves of a nonlinear ordinary differential equation (ODE), we are able to improve the conclusions of the recent contribution [A. Constantin, Proc. Japan Acad. {\bf 86(A)} (2010), 41--44]. In this way, we establish a flexible uniqueness criterion for ODEs without Lipschitz-like nonlinearities.